A private equity firm invested a total of \1,000,000 divided between two portfolio compani… — GMAT

A private equity firm invested a total of $\$1,000,000$ divided between two portfolio companies, Company A and Company B. During the first year, the value of Company A increased by $x\%$ , while the value of Company B decreased by $x\%$ , where $x > 0$ . At the end of the first year, the value of Company A was $\$720,000$ , and the combined value of both companies was $\$1,040,000$ . If the value of Company B decreased by an additional $x\%$ during the second year relative to its value at the end of the first year, what was the value, in dollars, of Company B at the end of the second year?

Answer: 256000 dollars

Answer

The value of Company B at the end of the second year was 256,000 dollars.

By representing the initial investments as algebraic expressions involving the multiplier

k = \frac{x}{100}

and setting up the total initial capital equation

A_0 + B_0 = 1,000,000

, we obtain a quadratic equation in

k

. Solving

(5k-1)^2 = 0

yields

k = 0.2

(

x = 20\%

). Decreasing Company B's Year 1 value of

\$320,000

20\%

gives

\$256,000

Step-by-Step Solution

Find the value of Company B at the end of Year 1

Year 1 value of Company B =

1,040,000 - 720,000 = 320,000

dollars

The total combined value of both companies at the end of Year 1 is given as $1,040,000.

Formulate algebraic expressions for initial values using rate

k = \frac{x}{100}

A_0 = \frac{720,000}{1+k}

and

B_0 = \frac{320,000}{1-k}

Company A increased by

x\%

A_1 = A_0(1+k)

; Company B decreased by

x\%

B_1 = B_0(1-k)

Set up and simplify the quadratic equation for the combined initial investment

\frac{720,000}{1+k} + \frac{320,000}{1-k} = 1,000,000 \implies 25k^2 - 10k + 1 = 0

Dividing by

80,000

yields

\frac{9}{1+k} + \frac{4}{1-k} = 12.5

. Multiplying through by

2(1-k^2)

leads to

2[9(1-k) + 4(1+k)] = 25(1-k^2)

Solve for

k

and determine percentage

x

(5k - 1)^2 = 0 \implies k = 0.2

, so

x = 20\%

Factoring the perfect square quadratic expression gives a unique solution for

k

Compute Company B's value at the end of Year 2

320,000 \times (1 - 0.2) = 256,000

dollars

Company B's value decreases by an additional

20\%

of its Year 1 value (

320,000

Key Concept

Algebraic Modeling of Rational/Quadratic Equations from Multi-Step Percent Change Scenarios

Estimated Time:2m 30s

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